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1.3soln

# 1.3soln - Solutions to problems from section 1.3 4 Let u =...

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Solutions to problems from section 1.3 4. Let u = bracketleftbigg 3 2 bracketrightbigg and v = bracketleftbigg 2 - 1 bracketrightbigg . Display the following vectors using arrows on an xy -graph: u , v , - v , - 2v , u + v , u - v , and u - 2v . Solution: 6. Write a linear system which is equivalent to x 1 bracketleftbigg - 2 3 bracketrightbigg + x 2 bracketleftbigg 8 5 bracketrightbigg + x 3 bracketleftbigg 1 - 6 bracketrightbigg = bracketleftbigg 0 0 bracketrightbigg Solution: - 2 x 1 + 8 x 2 + x 3 = 0 , 3 x 1 + 5 x 2 - 6 x 3 = 0 . 10. Write a vector equation that is equivalent to the following linear sys- tem: 4 x 1 + x 2 + 3 x 3 = 9 , x 1 - 7 x 2 - 2 x 3 = 2 , 8 x 1 + 6 x 2 - 5 x 3 = 15 . Solution: x 1 4 1 8 + x 2 1 - 7 6 + x 3 3 - 2 - 5 = 9 2 15

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12. Is b a linear combination of a 1 , a 2 , and a 3 where a 1 = 1 - 2 2 , a 2 = 0 5 5 , a 3 = 2 0 8 , b = - 5 11 - 7 ? Solution: Answering this question is the same as answering whether or not the linear system with augmented matrix 1 0 2 - 5 - 2 5 0 11 2 5 8 - 7 is consistent. Let’s row reduce. 1 0 2 - 5 - 2 5 0 11 2 5 8 - 7 NR 2 = R 2 +2 R 1 -→ 1 0 2 - 5 0 5 4 1 2 5 8 - 7 NR 3 = R 3 - 2 R 1 -→ 1 0 2 - 5 0 5 4 1 0 5 4 3
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1.3soln - Solutions to problems from section 1.3 4 Let u =...

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